Date: 06/29/2003 at 14:50:42
From: Doctor Rob
Subject: Re: How can we find the square root of matrix A...
Thanks for writing to Ask Dr. Math, Alfredo.
The usual way of finding the square root of a real symmetric matrix
involves diagonalizing it. Find an invertible matrix U such that
[d1 0]
U*M*U^(-1) = [ ] = D.
[0 d2]
d1 and d2 are the eigenvalues of M. U is not unique, but any one that
works in the above equation will do. In the case at hand, it turns out
that d1 = 1 and d2 = 3, or vice versa. Then there are four square
roots of D:
[sqrt(d1} 0 ] [sqrt(d1} 0 ]
[ ], [ ],
[ 0 sqrt(d2)] [ 0 -sqrt(d2)]
[-sqrt(d1} 0 ] [-sqrt(d1} 0 ]
[ ], [ ].
[ 0 sqrt(d2)] [ 0 -sqrt(d2)]
(Be sure you understand why each of the four, when squared, gives D as
the answer.) Let R be any one of them. Then
U*M*U^(-1) = D = R^2,
M = U^(-1)*R^2*U = [U^(-1)*R*U]*[U^(-1)*R*U],
so U^(-1)*R*U is one of the four square roots of M.
Feel free to write again if I can help further.
- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/